Hanke9 Odd-num Sol 05
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CHAPTER 5 TIME SERIES AND THEIR COMPONENTS ANSWERS TO ODD-NUMBERED PROBLEMS AND CASES 1. The purpose of decomposing a time series variable is to observe its various elements in isolation. By doing so, insights into the causes of the variability of the series are frequently gained. A second important reason for isolating time series components is to facilitate the forecasting process. 3. The basic forces that affect and help explain the trend-cycle of a series are population growth, price inflation, technological change, and productivity increases. 5. Weather and the calendar year such as holidays affect the seasonal component. 7. a. & b. 1
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Transcript of Hanke9 Odd-num Sol 05
CHAPTER 5
TIME SERIES AND THEIR COMPONENTS
ANSWERS TO ODD-NUMBERED PROBLEMS AND CASES
1. The purpose of decomposing a time series variable is to observe its various elements in isolation. By doing so, insights into the causes of the variability of the series are
frequently gained. A second important reason for isolating time series components is to facilitate the forecasting process.
3. The basic forces that affect and help explain the trend-cycle of a series are
population growth, price inflation, technological change, and productivity increases.
5. Weather and the calendar year such as holidays affect the seasonal component.
7. a. & b.
c. = 9310 + 1795(19) = 43,415
d. Cyclical component might be indicated because of wavelike behavior ofobservations about fitted straight line. However, if there is a cyclical affect, it is very slight.
9. = TS = 850(1.12) = $952
11. All of the statements are correct except d.
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13. a. & b. Would use both the trend and seasonal indices to forecast although seasonal component is not strong in this example (see plot and seasonal indices below).
Fitted Trend Equation
Yt = 2268.0 + 22.1*t
Seasonal Indices
Period Index 1 0.969 2 1.026 3 1.000 4 1.005
Forecasts
Period ForecastQ3/1996 3305.39Q4/1996 3343.02
c. The forecast for third quarter is a bit low compared to ValueLine’s forecast (3,305 versus 3,340). The forecast for fourth quarter
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is a bit high compared to Value Line’s (3,343 versus 3,300). Additional plots associated with the decomposition follow.
15. a. Additive Model
Data LnSalesLength 77NMissing 0
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Fitted Trend Equation
Yt = 4.6462 + 0.0215*t
Seasonal Indices
Period Index 1 0.335 2 -0.018 3 -0.402 4 -0.637 5 -0.714 6 -0.571 7 -0.273 8 -0.001 9 0.470 10 0.723 11 0.747 12 0.342
b. Pronounced trend and seasonal components. Would use both to forecast.
c. & d. Forecasts
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Month Forecast of LnSales Forecast of SalesJun/2006 5.75297 315Jul/2006 6.07205 434Aug/2006 6.36535 581Sep/2006 6.85802 951Oct/2006 7.13248 1252Nov/2006 7.17779 1310Dec/2006 6.79477 893
e. Forecasts of Cavanaugh sales developed from additive decomposition are higher (for all months June 2006 through December 2006) than those developed from the multiplicative decomposition. Forecasts from multiplicative decomposition appear to be a little more consistent with recent behavior of Cavanaugh sales time series.
17. a.
Variation appears to be increasing with level. Multiplicative decomposition may be appropriate or additive decomposition with the
logarithms of demand.
b. Neither a multiplicative decomposition or an additive decomposition with a linear trend work well for this series. This time series is best modeled with other methods. The multiplicative decomposition is pictured below.
c. Seasonal Indices (Multiplicative Decomposition for Demand)
Period Index Period Index Period Index 1 0.947 5 1.004 9 1.045
2 0.950 6 1.007 10 0.982
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3 0.961 7 1.022 11 0.995 4 0.998 8 1.070 12 1.019
Demand tends to be relatively high in the late summer months.
d. Forecasts derived from a multiplicative decomposition of demand (see plot below).
Month Forecast
Oct/1996 171.2 Nov/1996 174.9
Dec/1996 180.5
19. a. Jan = 600
12.= 500
500(1.37) = 685 people estimated for Feb
b. = 140 + 5(t); t for Jan 2007 = 72. = 140 + 5(72) = 500
Jan = 500(1.20) = 600 Feb = (140 + 5(73))(1.37) = 692 Mar = (140 + 5(74))(1.00) = 510 Apr = (140 + 5(75))(0.33) = 170 May = (140 + 5(76))(0.47) = 244 Jun = (140 + 5(77))(1.25) = 656
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Jul = (140 + 5(78))(1.53) = 811 Aug = (140 + 5(79))(1.51) = 808 Sep = (140 + 5(80))(0.95) = 513 Oct = (140 + 5(81))(0.60) = 327 Nov = (140 + 5(82))(0.82) = 451 Dec = (140 + 5(83))(0.97) = 538
c. 5
23. 1289.73(2.847) = 3,671.86 25. Multiplicative Model
Data Employed MenLength 130NMissing 0
Fitted Trend Equation
Yt = 65355 + 72.7*t
Seasonal Indices
Month Index Month Index 1 0.981 7 1.019 2 0.985 8 1.014
3 0.990 9 1.002 4 0.995 10 1.004 5 1.002 11 0.999 6 1.014 12 0.995
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Forecasts
Month ForecastNov/2003 74791.4Dec/2003 74581.7Jan/2004 73607.8
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Feb/2004 73954.0Mar/2004 74393.4Apr/2004 74887.2May/2004 75454.0Jun/2004 76419.5Jul/2004 76894.1Aug/2004 76564.4Sep/2004 75757.2Oct/2004 76005.6
A multiplicative decomposition with a default linear trend is not quite right for thesedata. There is some curvature in the time series as the plot of the seasonally adjusteddata indicates. Not surprisingly, there is a strong seasonal component with employment relatively high in the summer and relatively low in the winter. In spite of the not quite linear trend, the forecasts seem reasonable.
27. Multiplicative Model
Data Wal-Mart SalesLength 56NMissing 0
Fitted Trend Equation
Yt = 1157 + 1088*t
Seasonal Indices
Quarter Index Q1 0.923
Q2 0.986 Q3 0.958 Q4 1.133
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Forecasts and Actuals
Quarter Forecast ActualsQ1/2004 58328 65443Q2/2004 63346 70466Q3/2004 62607 69261Q4/2004 75278 82819
Slight upward curvature in the Wal-Mart sales data so a linear trend is not quiteright. Not surprisingly, there is a strong seasonal component with 4th quartersales relatively high and 1st quarter sales relatively low. The forecasts for 2004 are uniformly below the actuals (primarily the result of the linear trend assumption)although the seasonal pattern is maintained. Here MPE = MAPE = 9.92%. Multiplicative decomposition better than additive decomposition but any decomposition that assumes a linear trend will not forecast sales for 2004 well.
CASE 5-1: THE SMALL ENGINE DOCTOR
1.
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3. SEASONAL FITTED VALUES AND
ADJUSTMENT FORECASTS, T*S MONTH FACTORS 2005 2006 2007
Jan 0.693 8.68 17.32 25.97 Feb 0.707 9.59 18.41 27.23 Mar 0.935 13.66 25.34 30.01 Apr 1.142 17.87 32.13 46.38 May 1.526 25.48 44.52 63.57 Jun 1.940 34.39 58.61 82.82 Jul 1.479 27.77 46.23 64.69 Aug 0.998 19.77 32.23 44.68 Sep 0.757 15.78 25.22 34.67 Oct 0.373 8.17 12.83 17.49 Nov 0.291 6.68 10.32 13.95 Dec 1.290 30.94 47.06 63.17
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5. Trend*Seasonality (T*S): MAD = 1.52Linear Trend Model: MAD = 9.87
CASE 5-2: MR. TUX
At last, John is able to deal directly with the strong seasonal effect in his monthly data. Students find it interesting that in addition to using these to forecast, John's banker wants them to justify variable loan payments.
To forecast using decomposition, students see that both the C and I components must be estimated. We like to emphasize that studying the C column in the computer printout is helpful, but that other study is needed to estimate the course of the economy over the next several months. The computer is not able to make such forecasts with accuracy, as anyone who follows economic news well knows.
Thinking about John’s efforts to balance his seasonal business to achieve a more uniform sales picture can generate a good class discussion. This is usually the goal of any business; examples such as boats/skis or bikes/skis illustrate this effort in many seasonal businesses. In fact, John Mosby put a great deal of effort into expanding his Seattle business in order to balance his seasonal effect. Along with his shirt making business, he has achieved a rather uniform monthly sales volume.
1. The two sentences might look something like this: A computer analysis of John Mosby's monthly sales data clearly shows the strong variation by month. I think we
are justified in letting him make variable monthly loan payments based on the seasonal indices shown in the computer printout.
3. Using the sales figures for January and February of 2005, to get “average” (100%) sales
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dollars, divide the actual sales by the corresponding seasonal index:
Jan: 71,043/.314 = 226,252 Feb: 152,930/.472 = 324,004
Now subtract the actual sales from these target values to get the sales necessary from the shirt making machine:
Jan: 226,252 - 71,043 = 155,209 Feb: 324,004 - 152,930 = 171,074
CASE 5-3: CONSUMER CREDIT COUNSELING
Both the trend and seasonal components are important. The trend explains about 34% percent of the total variance.
Multiplicative Model
Data ClientsLength 99
Fitted Trend Equation
Yt = 89.88 + 0.638*t
Seasonal Indices
Month Index 1 1.177
2 1.168 3 1.246 4 0.997 5 0.940 6 1.020 7 0.916 8 0.951 9 0.878 10 1.055 11 0.868 12 0.783
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The number of new clients tends to be relatively large during the first three months of the year.
Forecasts
Month ForecastApr/2003 153.207May/2003 145.121Jun/2003 158.062Jul/2003 142.440Aug/2003 148.560Sep/2003 137.749Oct/2003 166.161Nov/2003 137.261Dec/2003 124.277
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There is one, possibly two, large positive residuals (irregularities) at the beginning of the series but there are no significant residual autocorrelations.
CASE 5-4: MURPHY BROTHERS FURNITURE
1. An adequate model was found using Holt’s linear exponential smoothing.
Smoothing Constants
Alpha (level) 0.980Gamma (trend) 0.025
Accuracy Measures
MAPE 1.1MAD 76.2
MSD 11857.8
Forecasts
Month Forecast Jan/2002 8127.8 Feb/2002 8165.1
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Mar/2002 8202.4 Apr/2002 8239.7 May/2002 8277.0 Jun/2002 8314.2 Jul/2002 8351.5 Aug/2002 8388.8 Sep/2002 8426.1
3. Using the same procedure as in 2, the forecast for October, 2002 is 8609.2.
CASE 5-5: AAA WASHINGTON
1. An additive and a multiplicative decomposition perform equally well. The multiplicativedecomposition is shown below.
Multiplicative Model
Data CallsLength 60NMissing 0
Fitted Trend Equation
Yt = 21851 - 17.0437*t
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Seasonal Indices
Month Index 1 0.937 2 0.922 3 0.972 4 0.963 5 0.925 6 1.016 7 1.063 8 1.094 9 1.094
11 1.025 12 0.936
Accuracy Measures: MAPE 4 MAD 814 MSD 1276220
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CASE 5-6: ALOMEGA FOOD STORES
The sales data for the Alomega Food Stores case is subjected to a multiplicative decomposition procedure in this case. A trend line is first calculated with the actual data plotted around it (using MINITAB). Students can project this line into future months for sales forecasts, although, as the case suggests, accurate forecasts will not result: The MAPE using only the trend line is 28%.
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A plot of the seasonal indices from the MINITAB output is shown below.. Students can summarize the managerial benefits to Julie from studying these values. As noted in the case, the MAPE drops to 12% when the seasonal indices along with the trend are used.
Finally, a 12-month forecast is generated using both the trend line and the seasonal indices. The forecasts seem reasonable.
Month ForecastJan/2007 785348Feb/2007 326276Mar/2007 585307Apr/2007 391827May/2007 558299Jun/2007 453257Jul/2007 520615Aug/2007 319029Sep/2007 614997Oct/2007 394599Nov/2007 377580Dec/2007 235312
There are no significant residual autocorrelations.
Although more a management concern than a forecasting one, the attitude of Jackson Tilson in the case might generate a discussion that ties the computer assisted forecasting process into the real-life personalities of business associates. Although increasingly unlikely in the business setting, there are still those whose backgrounds do not include familiarity with computer based data analysis. Students whose careers will be spent in business might benefit from a discussion of the human element in the management process.
CASE 5-7: SURTIDO COOKIES
1. Multiplicative Model
Data SurtidoSales
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Length 41NMissing 0
Fitted Trend Equation
Yt = 907625 + 4736*t
Seasonal Indices
Month Index 1 0.696 2 0.546 3 0.517 4 0.678 5 0.658 6 0.615 7 0.716
8 0.567 9 1.527 10 1.664
11 1.988 12 1.829
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Month ForecastJun/2003 680763Jul/2003 795362Aug/2003 633209Sep/2003 1710846Oct/2003 1872289Nov/2003 2246745
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Dec/2003 2076183
3. The residual autocorrelation function is shown below. There are no significantresidual autocorrelations.
The multiplicative decomposition adequately accounts for the trend and seasonality in the data. The forecasts are very reasonable. Jame should change his thinkingabout the value of decomposition analysis.
CASE 5-8: SOUTHWEST MEDICAL CENTER
1. Decomposition of a time series involves isolating the underlying componentsthat make up the time series. These components are the trend or trend/cycle(long term growth or decline), the seasonal (consistent within year variationtypically related to the calendar) and the irregular (unexplained variation).
3. There is a distinct upward trend in total visits. The seasonal indices show that visits in December (4th month of fiscal year) tend to be relatively low and visitsin July (11th month of fiscal year) tend to be relatively high.
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